# Fluctuation-dissipation theorem

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Thermal radiation between solids is often treated as a surface phenomenon and analyzed using ray optics with the assistance of the concept of emissivity, reflectivity and absorptivity of the surfaces. Radiation heat transfer inside a participating medium is traditionally dealt with by the radiative transfer equation (RTE), considering emission, absorption, and scattering [1]. However, these phenomenological approaches do not fully account for the origin of thermal emission and break down when wave interference and diffraction become important. According to the fluctuation-dissipation theorem, thermal emission is originated from the fluctuating currents induced by the random thermal motion of charges, known as thermally induced dipoles. The fluctuational electrodynamics, pioneered by Rytov and co-workers in 1950s, combines the fluctuation-dissipation theorem with Maxwell's equations to describe the propagation of thermal radiation, i.e., thermally radiated electromagnetic waves, and its interaction with matter in both the near and far fields [2].

The random thermal fluctuations of charges, such as electrons in metals or ions in polar crystals, generate fluctuating electric currents that radiate an electromagnetic field. Thus, the fluctuating electric density $\textbf j(\textbf x,t)\,,$ or $\textbf j(\textbf x,\omega )$ in the frequency domain, inside the medium can be considered as an external random source in the Maxwell equations, making the Maxwell equations stochastic in nature. The key issue is then to know the statistical properties of these random sources, which is given by the fluctuation-dissipation theorem (FDT). FDT allows us to derive the ensemble average of the spatial correlation function of the current density [3]:

$\left\langle j_{m} (x',\omega )j_{n} ^{*} (x'',\omega ')\right\rangle =\frac{4}{\pi } \omega \varepsilon _{0} Im(\varepsilon (\omega ))\delta _{mn} \delta (x'-x'')\Theta (\omega ,T)\delta (\omega -\omega ')$

(1)

where $<\, >$ represents ensemble averaging, and * denotes the complex conjugate. In Eq. (1), $\varepsilon _{0}$ is the electrical permittivity of the free space, $\, j_{m}$ and jn(m,n = 1,2,or3) stands for the x, y, or z component of j, δmn is the Kronecker delta, and δ(ω − ω') is the Dirac delta function. Θ(ω,T) is the mean energy of a Planck oscillator at the frequency ω in thermal equilibrium at temperature T and is given by $\Theta (\omega ,T)=\hbar \omega /[\exp (\hbar \omega /k_{{\rm B}} T)-1]$, where ${ \hbar }$ is the Planck constant divided by , and kB is the Boltzmann constant. Since only positive values of frequencies are considered here, a factor of 4 has been included in Eq. (1) to consistently use the conventional definitions of the spectral energy density and the Poynting vector [4].

## References

[1] Howell, J. R., Siegel, R, and Menguc, M. P., 2010, Thermal Radiation Heat Transfer, 5th edn., CRC Press/Taylor & Francis Group, New York.

[2] Rytov, S. M., Krastov, Yu. A., and Tatarskii, V. I., 1987, Principles of Statistical Radiophysics, Vol. 3, Springer-Verlag, New York.

[3] Joulain, K., Mulet, J.-P., Marquier, F., Carminati, R., and Greffet, J.-J., 2005, “Surface Electromagnetic Waves Thermally Excited: Radiative Heat Transfer, Coherence Properties and Casimir Forces Revisited in the Near-Field,” Surface Science Report, 57, pp. 59-112.

[4] Fu, C. J., and Zhang, Z. M., 2006, "Nanoscale Radiation Heat Transfer for Silicon at Different Doping Levels," International Journal of Heat and Mass Transfer, 49, pp. 1703-1718.